January 31

Q1

Find
$2^{220} \mod 221$
Conclude that 221 is composite.

Q2

Show that
$a^{48} \equiv 1 \mod 221$
for all a relatively prime to 221.

Q3

Solve the congruence
$x^7\equiv 3 \mod 21.$

Q4

Show that the congruence
$x^2=a \mod 21$
can have 0, 1, 2 or 4 solutions depending on a.